Construction to double the area of a circle?
Given a circle with a diameter d and area A, what are the steps in order to construct a circle with area 2A as well as one with area 5A? Thanks a lot!
You need to construct √2 and √5. You can do this by a classic construction method for finding the geometric mean √(AB), given A and B. So, for example, if A = B = 1, you have √2, and for A = 1 and B = 5, you have √5. Draw a semicircle of radius (1/2)(A+B), and then draw a perpendicular to the line from a point which is either A or B from either end of the diameter. The length of this perpendicular to the intersection with the semicircle is the length you seek, √(AB).
You need to construct √2 and √5. You can do this by a classic construction method for finding the geometric mean √(AB), given A and B. So, for example, if A = B = 1, you have √2, and for A = 1 and B = 5, you have √5. Draw a semicircle of radius (1/2)(A+B), and then draw a perpendicular to the line from a point which is either A or B from either end of the diameter. The length of this perpendicular to the intersection with the semicircle is the length you seek, √(AB).
References :
http://planetmath.org/encyclopedia/CompassAndStraightedgeConstructionOfGeometricMean.html
http://en.wikipedia.org/wiki/Semicircle
Another way of constructing square root of 2 and square root of 5 is to use right triangles.
Construct rectangles 1 x 1 and 1 x 2.
That is easily accomplished by constructing perpendicular lines
and marking off either 1 or 2 units from the point of intersection.
(You can draw a perpendicular by bisecting a straight angle.)
Then draw the diagonals of those rectangles and you have the desired lengths.
References :